Lauren plans to deposit $6000 into a bank account at the beginning of next month and $250/month into the same account at the end of that month and at the end of each subsequent month for the next 6 years. If her bank pays interest at a rate of 2%/year compounded monthly, how much will Lauren have in her account at the end of 6 years? (Assume she makes no withdrawals during the 6-year period. Round your answer to the nearest cent.)





To solve this, we'll break the problem into two parts:

1. **The initial deposit of $6000**: This deposit earns interest for the entire 6 years.
2. **Monthly deposits of $250**: These are made at the end of each month for 6 years and also earn interest, but for varying lengths of time depending on when they are deposited.

### Key Information:
- Interest rate: 2% per year compounded monthly
- Initial deposit: $6000
- Monthly deposit: $250
- Duration: 6 years (72 months)

### 1. **Future Value of the Initial Deposit**

The future value of a lump-sum investment (like the $6000 deposit) compounded monthly can be calculated using the compound interest formula:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- $A$ = future value of the investment
- $P$ = initial deposit (6000)
- $r$ = annual interest rate (2% or 0.02)
- $n$ = number of times the interest is compounded per year (12 for monthly)
- $t$ = number of years (6)

Substituting the known values:

\[
A = 6000 \left(1 + \frac{0.02}{12}\right)^{12 \times 6}
\]

### 2. **Future Value of the Monthly Deposits**

The future value of a series of monthly deposits (annuity) can be calculated using the future value of an ordinary annuity formula:

\[
A = P \times \frac{\left(1 + \frac{r}{n}\right)^{nt} - 1}{\frac{r}{n}}
\]

Where:
- $A$ = future value of the annuity
- $P$ = monthly deposit (250)
- $r$ = annual interest rate (2% or 0.02)
- $n$ = number of times interest is compounded per year (12)
- $t$ = number of years (6)

Substituting the values:

\[
A = 250 \times \frac{\left(1 + \frac{0.02}{12}\right)^{12 \times 6} - 1}{\frac{0.02}{12}}
\]

Now, let’s calculate the two components.

Lauren will have approximately **$25,871.94** in her account at the end of 6 years.

Would you like more details on any of the calculations, or do you have any questions?

Here are 5 related questions you might find interesting:
1. What would the final amount be if the interest rate was 3% compounded monthly?
2. How would the total change if she made her monthly deposits at the beginning of each month instead of the end?
3. What if she deposited $5000 initially instead of $6000?
4. How much total interest did she earn over the 6 years?
5. How does compound interest differ from simple interest in this scenario?

**Tip**: The longer the money is left in an account that earns compound interest, the more the interest compounds on itself, resulting in exponential growth over time.

Lauren plans to deposit $6000 into a bank account at the beginning of next month and $250/month into the same account at the end of that month and at the end of each subsequent month for the next 6 years. If her bank pays interest at a rate of 2%/year compounded monthly, how much will Lauren have in her account at the end of 6 years? (Assume she makes no withdrawals during the 6-year period. Round your answer to the nearest cent.)

Learn how to calculate the future value of a $6000 initial deposit and $250 monthly contributions over 6 years, with a 2% interest rate compounded monthly. Includes detailed breakdowns using the compound interest and annuity formulas.

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